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J Heuristics (2010) 16: 511–535 DOI 10.1007/s10732-009-9108-4

Multi-objective redundancy allocation optimization using a variable neighborhood search algorithm Yun-Chia Liang · Min-Hua Lo

Received: 24 November 2007 / Revised: 17 May 2009 / Accepted: 15 June 2009 / Published online: 27 June 2009 © Springer Science+Business Media, LLC 2009

Abstract A variable neighborhood search (VNS) algorithm has been developed to solve the multiple objective redundancy allocation problems (MORAP). The single objective RAP is to select the proper combination and redundancy levels of components to meet system level constraints, and to optimize the specified objective function. In practice, the need to consider two or more conflicting objectives simultaneously increases nowadays in order to assure managers or designers’ demand. Amongst all system level objectives, maximizing system reliability is the most studied and important one, while system weight or system cost minimization are two other popular objectives to consider. According to the authors’ experience, VNS has successfully solved the single objective RAP (Liang and Chen, Reliab. Eng. Syst. Saf. 92:323–331, 2007; Liang et al., IMA J. Manag. Math. 18:135–155, 2007). Therefore, this study aims at extending the single objective VNS algorithm to a multiple objective version for solving multiple objective redundancy allocation problems. A new selection strategy of base solutions that balances the intensity and diversity of the approximated Pareto front is introduced. The performance of the proposed multiobjective VNS algorithm (MOVNS) is verified by testing on three sets of complex instances with 5, 14 and 14 subsystems respectively. While comparing to the leading heuristics in the literature, the results show that MOVNS is able to generate more non-dominated solutions in a very efficient manner, and performs competitively in all performance measure categories. In other words, computational results reveal the advantages and benefits of VNS on solving multi-objective RAP. Keywords Variable neighborhood search · Redundancy allocation problem · Multi-objective Y.-C. Liang () · M.-H. Lo Department of Industrial Engineering and Management, Yuan Ze University, No 135 Yuan-Tung Road, Chung-Li, Taoyuan County, 320 Taiwan, ROC e-mail: [email protected]

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1 Introduction A series system of s independent k-out-of-n : G subsystems is considered the most studied configuration among the redundancy allocation problem (RAP) variations. The single objective RAP belongs to NP-hard problems (Chern 1992) and has been solved using various mathematical programming and other optimization approaches. Exact methods to the RAP include dynamic programming (Fyffe et al. 1968; Nakagawa and Miyazaki 1981; Li 1996; Onishi et al. 2007), integer programming (Ghare and Taylor 1969; Bulfin and Liu 1985; Misra and Sharma 1991; Coit and Liu 2000), and mixed-integer and nonlinear programming (Tillman et al. 1977). Owing to the computational expense of exact methods grows exponentially with the increase of problem sizes, meta-heuristics have become popular methods for solving the RAP in the past decade. Heuristic methods to the single objective RAP consist of Ant Colony Optimization (ACO) (Liang 2001; Huang et al. 2002; Shelokar et al. 2002; Huang 2003; Liang and Smith 2004; Zhao et al. 2007), Genetic Algorithm (GA) (Coit and Smith, 1996a, 1996b), Simulated Annealing (SA) (Ravi et al. 1997), Tabu Search (TS) (Huang et al. 2002; Kulturel-Konak et al. 2003), hybrid Neural Network (NN) and GA (Coit and Smith 1996c), hybrid ACO with TS (Huang 2003), ACO with Degraded Ceiling (Nahas et al. 2007), Great Deluge Algorithm (GDA) (Ravi 2004), and Variable Neighborhood Search (VNS) and its variations (Chen 2005; Liang and Wu 2005; Liang and Chen 2007; Liang et al. 2007), etc. The single objective RAP is to select the proper combination and redundancy levels of components to meet system level constraints, and to optimize the specified objective function. Most single objective RAP studies focus on maximizing system reliability or minimizing system cost. However, in practice, a decision maker or a system designer has to consider multiple objectives at the same time. In addition to reliability and cost, weight and volume are also important characteristics considered in the system level. Therefore, multiple objective functions become an essential aspect in the reliability design of engineering system. Besides, the studies on the multi-objective redundancy allocation problem are still limited. For example, Elegbede and Adjallah (2003) developed a GA for a repairable system. The authors transformed multi-objective formulation to a single-objective optimization problem and an exterior penalty function technique is employed. Sasaki and Gen (2003) proposed a hybridized Genetic Algorithm with a new chromosome representation for a fuzzy multi-objective problem. Taboada et al. (2007) proposed a datamining technique to truncate the set of non-dominated solutions. Salazar et al. (2006) developed an NSGA-II to solve three types of multi-objective reliability optimization problems. Kulturel-Konak et al. (2008) transformed a single-objective problem to a multi-objective function and solve it by a Tabu Search Algorithm. Variable Neighborhood Search (VNS) is one of the latest meta-heuristics and it was original proposed by Mladenovi´c (1995). Hansen and Mladenovi´c (2003) provide comprehensive surveys of state-of-the-art development of VNS and its variations. The single objective VNS algorithm employ a set of neighborhood search methods systematically to find the local optimum in each neighborhood iteratively and hopefully the search to the end can move toward the global optimum. Single objective VNS and its variations have been successfully applied to diverse com-

Multi-objective redundancy allocation optimization using a variable

513

binatorial optimization problems, e.g. graph coloring (Avanthay et al. 2003), multisource problem (Brimberg et al. 2000), P -median problem (Hansen and Mladenovi´c 1997), clustering (Hansen and Mladenovi´c 2002), minimum spanning tree problem (Ribeiro and Souza 2002), vehicle routing problem (Kytöjoki et al. 2007), and redundancy allocation problem (Liang and Wu 2005; Liang and Chen 2007; Liang et al. 2007), etc. Recently, the successful experience of VNS on single objective optimization problems has been applied to multi-objective problems. For example, Geiger (2004) and Gagné et al. (2005) proposed a randomized VNS algorithm and a hybrid tabu-VNS algorithm to solve scheduling problems, respectively. Stummer and Sun (2005) developed a VNS algorithm to solve capital investment planning problems. Therefore, in this study, the authors inherit their successful experience on single objective RAP by VNS to propose a Multi-Objective VNS (MOVNS) algorithm for multi-objective RAP. Three types of multi-objective RAP will be considered. The remaining paper is organized as follows: the problem definition is described in Sect. 2, and Sect. 3 introduces the proposed MOVNS algorithm. Performance measures, benchmark problems and computational results are discussed in Sect. 4. Finally, Sect. 5 provides the concluding remarks. 2 Problem definition As mentioned in Sect. 1, system reliability maximization is the most studied objective in single objective RAP. Therefore, three problems that each optimizes system reliability with another popular objective—system cost or system weight are discussed in this section. The mathematical models including the objective functions and constraints, and original resource of these three multi-objective redundancy allocation problems (MORAP) are provided as follows. Problem I Maximize Rs =

s    1 − (1 − Ri )xi

(1)

i=1

Minimize Cs =

s 

  Ci xi + exp(qi xi )

(2)

i=1

subject to the constraints s 

pi xi2 ≤ P ,

(3)

  Wi xi exp(qi xi ) ≤ W.

(4)

i=1 s  i=1

The first problem was proposed by Tillman et al. (1985). In Problem I, the objective is to determine the number of redundancies (xi ) at each stage i while there is only one component option at each stage i. Therefore, the system configuration can

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be considered as a series-parallel system with single component alternative in each subsystem. Two system objectives are optimized in this formulation: system reliability (Rs ) and system cost (Cs ). Equation (3) represents the constraint (P ) considering both weight and volume of components given that pi denotes the product of weight and volume per component. Equation (4) is the weight constraint (W ). Note that both (2) and (4) consider the additional cost (Ci × exp(qi xi )) and the multiplicative factor exp(qi xi ) respectively, due to the interconnecting parallel components. Problem II Maximize Rs =

s 

Ri (yi |ki )

(5)

Ci (yi )

(6)

i=1

Minimize Cs =

s  i=1

subject to the constraints s 

Wi (yi ) ≤ W,

(7)

i=1

ki ≤

ai 

xij ≤ nmax

∀i = 1, 2, . . . , s.

(8)

j =1

Problem II considers a series system that consists of s k-out-of-n: G subsystems. The goal of Problem II is to determine the optimal arrangement and redundancy levels of components which satisfies system weight constraint (W ) and optimizes the system reliability (Rs ) and system cost (Cs ) objectives simultaneously. Both system cost and system weight are calculated by the linear combination of component cost and component weight respectively, i.e., no interconnecting effect is considered in this model. Different from Problem I, each subsystem has multiple component choices, and the mix of component types in each subsystem is allowed in Problem II. Therefore, yi (= (xi1 , . . . , xiai )) denotes an ordered set of xij that represents the quantity of component type j used in subsystem i, and ai is the number of available component options in subsystem i. Equation (8) indicates that the total number of components a used in subsystem i, i.e., j i=1 xij , has to lie within the range of ki and nmax where ki denotes the minimum number of components in parallel required for subsystem i to function and nmax is the pre-determined maximum number of components allowed in parallel. Problem III Maximize Rs =

s 

Ri (yi |ki )

(9)

i=1

Minimize Ws =

s  i=1

Wi (yi )

(10)

Multi-objective redundancy allocation optimization using a variable

515

subject to the constraints s 

Ci (yi ) ≤ C,

(11)

i=1

ki ≤

ai 

xij ≤ nmax

∀i = 1, 2, . . . , s.

(12)

j =1

Problem III is an intuitive extension from Problem II by switching the cost objective and the weight constraint. That is Problem III aims at optimizing system reliability and system weight (Ws ) at the same time while has to satisfy the cost constraint. In addition, (12) equivalent to (8) is also provided to limit the redundancy level in each subsystem.

3 Multi-objective variable neighborhood search algorithm (MOVNS) Multi-Objective Variable Neighborhood Search (MOVNS) intends to extend the single-objective VNS algorithm to solve the multiple objective optimization problems. The main idea of MOVNS different from its single-objective VNS variations includes random selection of neighborhoods, a new selection strategy for the base solution, and the use of the neighboring solutions. The pseudo code of MOVNS is illustrated in Fig. 1. The MOVNS algorithm starts from generating a feasible initial solution randomly. This initial solution is also the first and only member in the starting approximated Pareto front. Then, a set of neighborhoods has to be defined properly. The search loop begins with the selection of base solution from the current approximated Pareto front. A neighborhood is then randomly selected with equal probability from the pool of the pre-defined neighborhood structures. Two search operators—shaking and neighborhood search are performed consecutively

Fig. 1 Procedure of the multi-objective variable neighborhood search (MOVNS) algorithm

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thereafter. Unlike in the single-objective VNS algorithm, only the best neighboring solution is considered to update the base solution, in MOVNS, all neighboring solutions will be adopted to update the approximated Pareto front. The search procedure continues until the pre-specified stopping criterion is reached. The following sections will discuss the steps of the proposed multi-objective variable neighborhood search algorithm in more details by starting with the initial solution generation. 3.1 Initial solution generation Random generation is the most common approach to construct an initial solution in meta-heuristics. When dealing with constrained optimization problems such as RAP, to generate a proper initial solution plays an important role in algorithmic efficiency. In this study, only a feasible initial solution is allowed. For all three types of problems, the number of initial components (ni ) is generated from a uniform distribution in the range of [LB, UB] for each subsystem i. In the formulation of Problem I, there exists merely one component option in each subsystem (stage), so ni units of identical component will be chosen at each stage. As for Problems II and III, multiple component choices are available in each subsystem, so the ni components will then be selected from all possible component alternatives randomly, i.e., each component type has the equal opportunity to be picked. 3.2 Update of the approximated Pareto front The idea of optimality in single-objective optimization problems is not directly applicable to multi-objective optimization problems, since no single global optimum can be found while considering all the objectives simultaneously. For example, in Problem I, if a solution y 1 has larger system reliability but a smaller system cost than the ones of another solution y 2 , then y 1 and y 2 are not superior to each other, i.e., y 1 and y 2 are non-dominated to each other. On the other hand, a solution y 1 dominates y 2 if y 1 is better than y 2 in one of the objectives, say system reliability, and performs equally or better in another objective, say system cost. For those solutions that cannot be dominated by any other solutions found in the search space are called non-dominated solutions. Therefore, the optimal solutions in a multi-objective optimization problem usually consist of a set of non-dominated solutions, i.e., approximated Pareto front. The formal definition of the Pareto optimality and related terms can be found in Salazar et al. (2006). In MOVNS, the approximated Pareto front starts with a solo member, i.e., the initial solution. The front is then updated constantly during the neighborhood search. All neighboring solutions are checked while updating the front. Those solutions dominated by the new candidate will be deleted from the approximated Pareto front, and the candidates not being dominated by any members in the front are admitted to enter the non-dominated set. The final approximated Pareto front is reported in the end of search procedure as the final solution is recorded in single-objective problem.

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Fig. 2 An example of type 1 neighborhood

3.3 Neighborhood structures Neighborhood structures in VNS and its variations determine how the search space will be explored efficiently. For complex problems such as RAP in this study, the number of neighborhood structures is usually limited. Therefore, two types of neighborhoods are employed. In addition, to save the computational expense, only a randomly chosen neighborhood is performed at each iteration in MOVNS. In the design of neighborhood structure for RAP, the idea of “empty” component is adapted. Namely an existing component can be replaced by an “empty” component or one extra “empty” slot is considered for changing to other available component options. For Problem I, neighborhood type 1 can increase or reduce the number of redundant components by one at each stage; for Problems II and III, the first type of neighborhood simultaneously replaces one type of existing component with a different type of component options in the same subsystem. Figure 2 illustrates an example of neighborhood type 1 for Problems II or III. Assume in the subsystem demonstrated, three component types (I, II and III) are available for selection, and the subsystem currently has two components I and II in parallel as shown in Fig. 2(a). The “shaded” component from Figs. 2(b) to 2(j) reveals the newly changed component. For instance, component type I is replaced by an “empty” component in Fig. 2(b), while an “empty” position is replaced by component type III in Fig. 2(j). Total of nine possible neighboring solutions are illustrated from Figs. 2(b) to 2(j). By the design of such neighborhood structure, the number of components can be increased or reduced by one or maintain at the same level. Also all available component options are considered. The second type of neighborhood structure involves the change of two components at the same time. For Problem I, type 2 neighbourhood is employed only when the number of components in a subsystem is greater than or equal to 3 since the neighborhood will reduced the number of components by two. As for Problems II and III, two existing components are simultaneously replaced by two different types of component options in the same subsystem. An example of neighborhood type 2 is illustrated in Fig. 3. Assume the subsystem presently has three components—types I, II, and III each as demonstrated in Fig. 3(a). All the possible neighboring solutions considering the changes in the first two positions (marked by the “shaded” components) are shown in Figs. 3(b) to 3(j). For instance, Fig. 3(c) shows that the component type I in the first position is replaced by an “empty” component, i.e. component “0”, and the component type II in the second position is replaced by component type I at the

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Y.-C. Liang, M.-H. Lo

Fig. 3 An example of type 2 neighborhood

same time. Note that only the feasible neighboring solutions are considered in the neighborhood, i.e., the solutions have to satisfy ki and nmax requirements. 3.4 Selection strategy of the base solution In a single-objective VNS algorithm, the base solution starting from the initial solution is updated through the search process. However, the selection of a base solution at each iteration in the proposed MOVNS algorithm is not as intuitive as its relative in single-objective but at the same time it also plays a key role in guiding the search direction. A special selection strategy, therefore, is developed here. The base solution in MOVNS is selected from the set of non-dominated solutions, i.e., the approximated Pareto front. To balance two important characteristics of the approximated Pareto front—diversity and intensity, the selection strategy tries to lead the search in three different directions. In the first direction, all non-dominated solutions are ranked in the descending order of objective 1—system reliability. Then starting from the top of the list, the first solution that has not been explored by any neighborhood will be selected as the base solution for the following search steps. Similarly, in the second direction, all non-dominated solutions are ranked in the ascending order of objective 2—either system cost or system weight. And the first unvisited non-dominated solution on the top of the list is chosen. Since the first two directions focus on the two extreme regions—the best of objective 1 and the best of objective 2 respectively, the third direction is then to pick an unexplored non-dominated solution from the set randomly. Therefore, whenever a base solution needs to be determined, one of these three search directions will be chosen randomly with uniform distribution, and then a base solution will be selected accordingly. In addition, once a non-dominated solution is explored by the neighborhood search and still stays in the approximated Pareto front thereafter, it will be marked as a visited solution. Thus, the visited solutions will be excluded from the selection of the base one. If all members in the approximated Pareto front are marked as visited before reaching the stopping criterion of MOVNS, then all marks will be reset and the selection procedure can start over again. Figure 4 illustrates three search directions on the approximated Pareto front.

Multi-objective redundancy allocation optimization using a variable

519

Fig. 4 Illustration of the selection strategy in MOVNS

3.5 Shaking The main purpose of the shaking operation, the last operation in the main search procedure, is to offer a stochastic mechanism for the selection of the new base solution and provides a better opportunity to escape from local optima. Shaking is performed after a base solution is selected and one of the neighbourhood structures defined in Sect. 3.3 is randomly chosen with equal probability. It will randomly generate a neighbouring solution of the current base solution using the selected neighbourhood, and this new solution will be employed for the following neighbourhood search. For example, as shown in Fig. 2(a), the base solution owns two components— component I and II each in parallel. If type 1 neighbourhood is employed, the outcome of the shaking operation will be one of the 9 neighbouring solutions illustrated from Figs. 2(b) to 2(j).

4 Test problems and results The proposed MOVNS algorithm is coded in Borland C++ Builder 6.0 and is run using an Intel Pentium IV 3.0 GHz PC with 1 GB RAM. All computations use real float point precision without rounding or truncating values. The system reliabilities of the final solutions are rounded to six digits behind the decimal point in order to compare with results in the literature. The number of digits behind the decimal point used in other methods is truly referred to each reference. Additionally, the number of initial components in each subsystem is randomly chosen within the range of [1, 2]. That is to generate a feasible initial solution; no more than two components will be used in each subsystem. The stopping criterion of MOVNS and the competing algorithms are described in the corresponding sections. Meanwhile, the stopping criterion of the reference (approximated) Pareto front (generated for comparison purpose) is ten times of number of evaluations in MOVNS, and the reference front is collected from the set of non-dominated points over ten runs.

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4.1 Performance measures As mentioned in Sect. 3, the concept of the single objective optimization problem does not apply directly to the multiple objective optimization problem. Therefore, four measures are selected to evaluate the performance of the multiple objective algorithms. The first two measures—hit ratio and accuracy ratio both calculate the percentage of reference non-dominated solutions found but used different denominators. In the hit ratio measure, the denominator is the number of points in the approximated Pareto front obtained by the competing algorithms while the number of nondenominated points in the reference Pareto front is employed as the denominator in the accuracy ratio. Other two measures, GD and D1R evaluate the distance between the approximated Pareto front and the reference Pareto front. GD considers the distance from the view point of the approximated Pareto front while D1R calculates the distance from the reference Pareto front. Therefore, the hit ratio and the accuracy ratio are the larger the better, and the GD and D1R values are the smaller the better. The formal definitions of these four measures are listed as follows: • Hit Ratio (Zitzler et al. 2003)  N 1 i=1 ei , where ei = Hit Ratio = |Yknow | 0

if a point in Yknow ∈ Ytrue otherwise

• Accuracy Ratio (Zitzler et al. 2003)  N 1 if a point in Yknow ∈ Ytrue i=1 ei , where ei = Accuracy Ratio = |Ytrue | 0 otherwise

(13)

(14)

• GD (Zitzler et al. 2003) ⎛ GD =

|Yknow |

di =

|Y know |

1



⎞1/2 min{di2 |Ytrue ∈ Yknow }⎠

(15)

i=1

(f1 (y ∗ ) − f1 (y))2 + · · · + (fN (y ∗ ) − fN (y))2

(16)

• D1R (Ishibuchi et al. 2003) 1

|Y true | 

min {di |Yknow ∈ Ytrue } |Ytrue | i=1 di = (f1 (y ∗ ) − f1 (y))2 + . . . + (fN (y ∗ ) − fN (y))2

D1R =

(17)

(18)

where ei is a binary variable denoting whether a solution belongs to the reference Pareto front, Ytrue represents the reference Pareto front, Yknow denotes the approximated Pareto front, |Y | is the number of solutions (points) in the set Y , y ∗ denotes the solution vector from the reference Pareto front, y is the solution vector from the approximated Pareto front, fi (.) represents the ith normalized objective function, and N is the number of objectives considered.

Multi-objective redundancy allocation optimization using a variable Table 1 Data for Problem I

521

i

Ri

pi

Ci

Wi

1

0.80

1

7

7

2

0.85

2

7

8

3

0.90

3

5

8

4

0.65

4

9

6

5

0.75

2

4

9

Fig. 5 (R, C) approximated Pareto front for Problem I

4.2 Problem I The first test problem was initially published by Tillman et al. (1985) in the form of single-objective and Salazar et al. (2006) extended to a multi-objective problem. Salazar et al. also proposed an NSGA-II to solve the problem. The product of weight and volume constraint P is set to 110, and the weight constraint W is 200. The multiplicative factor qi in exponential function is set to 0.25 for all subsystems. The number of subsystems is five, and the component data is listed in Table 1. The (R, C) approximated Pareto fronts of both MOVNS and NSGA-II are shown in Fig. 5, and the details of all 21 non-dominated solutions are provided in Appendix A. MOVNS is able to find all the non-dominated solutions as provided by NSGA-II. Note that the approximated Pareto fronts obtained by both MOVNS and NSGA-II are also exactly equivalent to the reference Pareto front. Therefore, the hit ratio and the accuracy are both equal to one while the GD and D1R are both equal to zero for both MOVNS and NSGA-II. The stopping criterion of MOVNS is when the total number of evaluations reaches 9,000. On the other hand, the CPU time of MOVNS on Problem I is very close to zero second, i.e., MOVNS is able to find the superior approximated Pareto front in no time.

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Table 2 Component data of Problems II and III Subsystem

Component 1

i

Ri

Ci

Wi

Component 2 Ri

Ci

1

0.9

1

3

0.93

1

2

0.95

2

8

0.94

1

3

0.85

2

7

0.9

3

4

0.83

3

5

0.87

4

5

0.94

2

4

0.93

6

0.99

3

5

0.98

7

0.91

4

7

8

0.81

3

9

0.97

2

10

0.83

11 12

Component 3 Wi

Component 4

Ri

Ci

Wi

Ri

Ci

Wi

4

0.91

2

2

0.95

2

5

10

0.93

1

9

5

0.87

1

6

0.92

4

4

6

0.85

5

4

2

3

0.95

3

5

3

4

0.97

2

5

0.96

2

4

0.92

4

8

0.94

5

9

4

0.9

5

7

0.91

6

6

8

0.99

3

9

0.96

4

7

0.91

3

8

4

6

0.85

4

5

0.9

5

6

0.94

3

5

0.95

4

6

0.96

5

6

0.79

2

4

0.82

3

5

0.85

4

6

0.9

5

7

13

0.98

2

5

0.99

3

5

0.97

2

6

14

0.9

4

6

0.92

4

7

0.95

5

6

0.99

6

9

4.3 Problem II Problem II was originally proposed by Fyffe et al. (1968) in the form of singleobjective and Nakagawa and Miyazaki (1981) extends to a set of 33 instances with different weight constraints. Salazar et al. (2006) further extend to a multi-objective problem with the objectives of maximizing system reliability and minimizing system cost, given weight constraint W = 191 for a series-parallel system with 14 subsystems. Each subsystem has three or four component alternatives. ki is set to one and nmax is equal to 8 for all subsystems. The component data including cost, weight and reliability is shown in Table 2. Note Problem III also shares the same component data with Problem II, and the reference Pareto front has 102 non-dominated solutions. Salazar et al. (2006) proposed this test problem and used a Non-dominated Sorting Genetic Algorithm (NSGA-II) to solve it. The number of evaluations in NSGA-II is approximately 5,000,000 and the archive size is set to 50. The number of nondominated solutions in the final approximated Pareto front by NSGA-II is 49 as reported in Salazar et al. (2006), and 10 of them fall on the reference Pareto front. To conduct a fair comparison with NSGA-II, MOVNS was executed in two different settings of archive size—50 or unbounded, and both employed the same number of evaluations 5,000,000 as NSGA-II in the literature. The performance measures of bounded archive and unbounded archive MOVNS, and NSGA-II are listed in Table 3. The MOVNS with bounded archive size consists of 50 points and 41 of them hit the reference Pareto front; therefore, the hit ratio and the accuracy ratio of MOVNS with the archive size of 50 are much higher than the ones by NSGA-II. And not surprisingly, MOVNS with archive size of 50 also outperforms NSGA-II in the GD measure, since more points found by MOVNS fall on the reference Pareto front than NSGA-II. The only measure NSGA-II performs competitively to MOVNS is D1R . As illustrated

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Table 3 Performance comparison of MOVNS, RVNS (Geiger 2004) and NSGA-II (Salazar et al. 2006) Algorithm

Archive size

MOVNS

Archive size = 50 Unbounded archive size

RVNS

Unbounded archive size

NSGA-II

Archive size = 50

Hit ratio

Accuracy ratio

GD

D1R

0.820000

0.401961

0.000018

0.032704

0.990196

0.990196

1.2E-07

1.2E-07

0.000000

0.000000

0.013475

0.119204

0.204080

0.098039

0.000729

0.014970

Fig. 6 (R, C) Approximated Pareto front for Problem II, archive size = 50 (part A—for non-dominated points with Rs ≥ 0.945 and Cs ≥ 80)

in Figs. 6 and 7, the approximated Pareto front obtained by NSGA-II spreads more evenly than the one by MOVNS. Thus, the D1R value of NSGA-II is slightly better than the one of MOVNS with archive size of 50. While the archive size of the MOVNS algorithm is relaxed to unbounded, i.e., the front will collect as many as non-dominated points as possible, the improvement on measures is significant as shown in Table 3. MOVNS in this case is able to find 102 points and 101 of them are located on the reference Pareto front. That explains the high values of both the hit ratio and the accuracy ratio value. And the GD and D1R values are also both close to zero as expected. The detailed solutions of both MOVNS with bounded and unbounded archives are provided in Appendix B for the future reference. In addition, the CPU time of MOVNS with archive size of 50 is 62.766 seconds while the MOVNS with unbounded archive needs 121.421 seconds. The higher CPU time consumption contributes to the update of the larger approximated Pareto front. Additionally, to show the competitive edge of the proposed MOVNS algorithm with the traditional multiobjective VNS algorithms, the randomized variable neighborhood search algorithm

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Y.-C. Liang, M.-H. Lo

Fig. 7 (R, C) Approximated Pareto front for Problem II, archive size = 50 (part B—for non-dominated points with Rs < 0.945 and Cs < 80)

(Geiger 2004), RVNS in short, is coded using the same environment, language, and the stopping criterion as MOVNS. The main differences between MOVNS and RVNS consist of the use of shaking operation and the selection strategy of the base solution. MOVNS applies a shaking operation before the employment of each neighborhood search while RVNS did not use any of the perturbation mechanism. RVNS chooses an “unvisited” solution randomly from the archive while a more evenly distributed selection strategy (as described in Sect. 3.4) is employed in MOVNS. From both Figs. 8 and 9, it is obvious that the approximated Pareto front obtained by RVNS is way behind the one by MOVNS. Also, the performance of MOVNS outperforms RVNS in all four measures. That verifies the newly developed selection strategy shows its merit on improving the efficiency and effectiveness of VNS algorithms. 4.4 Problem III One of the most popular single-objective RAP benchmark is the 33 system reliability maximization instances (proposed by Fyffe et al. 1968, and then extended by Nakagawa and Miyazaki 1981). In this benchmark set, the system cost constraint is set to 130, and the system weight constraint ranges from 159 to 191. Therefore, the objectives of Problem III are to maximize system reliability and minimize system weight given the system cost constraint of 130. Other problem settings such as component data, ki and nmax are as described in Sect. 4.3. The stopping criterion of MOVNS is when the number of evaluation reaches 7,000,000, and the number of non-dominate solutions in the reference Pareto front is 228. The number of non-dominated solutions obtained by the MOVNS algorithm is also 228 and the detailed results are shown in Appendix D. The CPU time of MOVNS is 190.969 seconds. Comparing with the

Multi-objective redundancy allocation optimization using a variable

525

Fig. 8 (R, C) Approximated Pareto front for Problem II (part A—for non-dominated points with Rs ≥ 0.945 and Cs ≥ 80)

Fig. 9 (R, C) Approximated Pareto front for Problem II (part B—for non-dominated points with Rs < 0.945 and Cs < 80)

reference Pareto front, the performance measures are as follows: D1R = 0.000007, GD = 0.000002, and Hit Ratio = Accuracy Ratio = 0.820175. To verify the performance of MOVNS, optima of the 33 single-objective instances provided by Onishi et al. (2007) are compared along with the results of some other famous algorithms in the literature such as GA (Coit and Smith 1996b), TS (Kulturel-

526

Y.-C. Liang, M.-H. Lo

Fig. 10 (R, W ) Approximated Pareto front for Problem III by MOVNS and the optimal solutions of 38 single objective instances

Konak et al. 2003), ACO (Liang and Smith 2004), IA (Chen and You 2005), Y&C (You and Chen 2005), VND (Liang and Wu 2005), VNS_RAP (Liang and Chen 2007), and ACO/DC (Nahas et al. 2007), etc. The authors find that optima of all 33 single objective instances are included in the non-dominated solutions of MOVNS which is illustrated in Fig. 10. To get a better view on the overlapped region between the 33 single objective optima and the non-dominated solutions, a zoom window is illustrated in Fig. 11. Again, MOVNS is able to provide a variety of solution alternatives. Table 4 summarizes the best performance of all competing algorithms in 33 singleobjective instances. The “shaded” region indicates that the optimum is found. Note that the best solution in each instance on VNS, GA, ACO, TS and VNS_RAP is obtained over 10 runs, on Y&C is from 20 trials, on ACO/DC is among 5 trials, and on IA and MOVNS are adapted from a single run. When comparing with those algorithms in the literature, MOVNS is the only algorithm that is able to find all 33 optimal solutions. In addition, Onishi et al. (2007) extended the system weight constraint to 198, 201, 202, 204 and 205. As shown in Table 5, the non-dominated solutions of MOVNS are also able to reach all the optimal solutions in the five extended instances. When comparing the computational expense, the number of total evaluations may provide a rough idea on how efficient an algorithm performs. Since many algorithms did not offer the number of evaluations in the literature, the approximate number of evaluations is obtained by calculating the product of the number of ants and the number of iterations in ACO, or the population size and the number of generations in GA, etc. Note the number of evaluations in local search is neglected if the literature did not mention particularly. Therefore the number of evaluations in a single run (trial) of each competing method is as follows: VND49,000, GA-48,040, ACO-100,000, TS-350,000, VNS_RAP-120,000, Y&C-9,820,

Multi-objective redundancy allocation optimization using a variable

527

Fig. 11 The zoom window of Fig. 10 (for 38 single objective optima and non-dominated points with 159 ≤ Cs ≤ 205 and 0.95 < Rs < 0.995)

Table 4 Comparison of computational results of 33 single-objective RAP instances

528

Y.-C. Liang, M.-H. Lo

Table 5 Comparison of computational results of five extended single-objective RAP instances

IA-360,120, and ACO/DC-150,000. The total number of evaluations in all 33 instances is then calculated by multiplying the number of test trails and the number of instances. Thus, the total number of evaluations in each method is VND-16,170,000, GA-15,853,200, ACO-33,000,000, TS-115,500,000, VNS_RAP-39,600,000, Y&C6,481,200, IA-11,883,960 and ACO/DC-24,750,000 while MOVNS needs only 1,013,158 to find all 33 optima. The total number of evaluations in MOVNS is obviously the smallest one among all competing algorithms. Note that the CPU time calculation per evaluation may differ over methods in the literature. For example, GA (Coit and Smith 1996b) had to recalculate the related system information for the entire system while TS (Kulturel-Konak et al. 2003) first proposed an approach that only needed to recalculate the related information of the subsystem being changed. For a complex system, this approach could save lots of computational effort while evaluating the objective value. Therefore, VNS_RAP (Liang and Chen 2007) and MOVNS in this study also employed the same approach as suggested by TS. Besides, it shows that MOVNS is capable of finding lots of solution alternatives including these singleobjective optima with a very reasonable computational effort.

5 Conclusions This paper proposes an MOVNS algorithm to solve the series-parallel system multiobjective redundancy allocation problems. VNS and its variations have been successfully applied to the single-objective RAP; however, this study should be the first application of VNS on the multiple-objective RAP according to authors’ knowledge. Two neighborhood structures are employed to explore the search space. And the most important of all, a new selection strategy is proposed to determine the base solution and properly leads the search direction. Three problems with the objectives of system reliability maximization and system weight/cost minimization are defined. In the first problem, the approximated Pareto front generated by MOVNS is the same as the ones by the competing algorithm NSGA-II and the reference Pareto front. For the second problem, when considering the archive size, MOVNS outperforms NSGA-II in the hit ratio, accuracy ratio, and GD value, and performs competitively in D1R . When relaxing the archive size to unlimited, MOVNS is superior to NSGA-II in all measures, and finds almost all non-dominated points in the reference Pareto front. For the last problem, the computational results of MOVNS reveal all the optimal solutions of 38 single objective RAP instances. The total number of evaluations of MOVNS needed is less than those of other competing algorithms in the literature. In general, this study has successfully proposed an MOVNS algorithm that is effective and efficient to solve

Multi-objective redundancy allocation optimization using a variable

529

the multi-objective RAPs. For the future research, because of the simple structure, it is believed that MOVNS can be easily applied to other multi-objective combinatorial optimization problems. The authors have also successfully employed the MOVNS algorithm to solve the multi-objective project portfolio optimization problem. In addition, the idea in the Pareto iterated local search metaheuristic (Geiger 2006) could be another interesting research direction for the further improvement in the current multi-objective VNS algorithms and its variations. Acknowledgements The authors would like to thank Dr. Daniel Salazar, Dr. Claudio M. Rocco and Dr. Blas J. Galván for sharing their results and provide valuable discussion. In addition, the authors also appreciate three reviewers and guest editors for their precious comments and suggestions.

Appendix A

Table 6 MOVNS results of Problem I Reliability

Cost

Weight

Reliability

Cost

Weight

1

0.904467

146.1250

192.4810

2

0.883248

143.1880

195.5350

12

0.634367

106.2560

120.5700

13

0.604159

100.3830

3

0.875291

135.8470

93.0881

171.1060

14

0.553812

97.6533

4

0.833610

95.1016

129.9740

143.6240

15

0.528639

96.7030

106.4760

5

0.812027

129.0370

193.4840

16

0.503466

90.8299

78.9942

6

0.802474

122.6320

152.7850

17

0.447525

88.1005

81.0076

7

0.764261

116.7590

125.3030

18

0.410231

85.3711

83.0211

8

0.729522

115.8090

136.6770

19

0.391584

84.4207

94.3957

9

0.697803

113.0790

136.6770

20

0.372938

78.5476

66.9137

10

0.694783

109.9360

109.1950

21

0.298350

73.0888

48.7930

11

0.664575

107.2060

109.1950

530

Y.-C. Liang, M.-H. Lo

Appendix B

Table 7 MOVNS results of Problem II (archive size = 50) Reliability

Cost

Weight

Reliability

Cost

Weight

Reliability

Cost

Weight

1

0.983349

118

191

2

0.982375

115

191

18

0.969547

93

191

19

0.929760

76

175

35

0.726312

57

130

36

0.713747

56

3

0.982042

114

191

20

0.907348

73

126

170

37

0.699328

55

4

0.981649

113

191

21

0.900439

125

72

169

38

0.685200

54

5

0.981205

112

191

22

125

0.892238

71

164

39

0.670621

53

6

0.980861

111

191

124

23

0.884111

70

157

40

0.646415

52

7

0.979916

109

121

191

24

0.875621

69

152

41

0.632662

51

8

0.979481

120

108

191

25

0.866776

68

152

42

0.617452

50

9

120

0.978600

106

191

26

0.849840

66

148

43

0.595165

49

117

10

0.977417

104

191

27

0.841255

65

148

44

0.582502

48

116

11

0.976683

103

191

28

0.828859

64

142

45

0.558966

47

113

12

0.976195

102

191

29

0.812115

63

141

46

0.511242

45

113

13

0.975523

101

191

30

0.799713

62

138

47

0.484959

44

110

14

0.974883

100

191

31

0.784146

61

135

48

0.462662

43

109

15

0.974184

99

191

32

0.768305

60

135

49

0.436959

42

108

16

0.973696

98

191

33

0.756572

59

131

50

0.412225

41

98

17

0.971690

95

191

34

0.741288

58

130

Multi-objective redundancy allocation optimization using a variable

531

Appendix C

Table 8 MOVNS results of Problem II (unbounded archive size) Reliability

Cost

Weight

Reliability

Cost

1

0.987503

135

191

2

0.987404

134

191

3

0.987206

133

4

0.987107

132

5

0.987008

6

Weight

Reliability

Cost

Weight

36

0.975036

100

191

71

0.841255

65

148

37

0.974294

99

191

72

0.828859

64

142

191

38

0.973807

98

191

73

0.812115

63

142

191

39

0.973027

97

191

74

0.799713

62

138

131

191

40

0.972469

96

191

75

0.784146

61

135

0.986811

130

191

41

0.971690

95

191

76

0.768305

60

134

7

0.986514

129

191

42

0.970909

94

191

77

0.756572

59

131

8

0.986349

128

191

43

0.969639

93

191

78

0.741288

58

130

9

0.986152

127

191

44

0.968474

92

191

79

0.726312

57

130

10

0.985954

126

191

45

0.966873

91

191

80

0.713747

56

126

11

0.985689

125

191

46

0.965706

90

191

81

0.699328

55

126

12

0.985492

124

191

47

0.964443

89

191

82

0.685200

54

125

13

0.985393

123

191

48

0.963182

88

191

83

0.670621

53

124

14

0.985196

122

191

49

0.961590

87

191

84

0.646415

52

121

15

0.984802

121

191

50

0.960332

86

191

85

0.632662

51

120

16

0.984461

120

191

51

0.957790

85

191

86

0.617452

50

120

17

0.984071

119

191

52

0.955342

84

191

87

0.595165

49

117

18

0.983677

118

191

53

0.954505

83

191

88

0.582502

48

116

19

0.983377

117

191

54

0.952020

82

187

89

0.558966

47

113

20

0.983180

116

191

55

0.948783

81

191

90

0.541315

46

114

21

0.982787

115

191

56

0.945692

80

184

91

0.511242

45

113

22

0.982295

114

191

57

0.941948

79

190

92

0.484959

44

110

23

0.981649

113

191

58

0.938759

78

180

93

0.462662

43

109

24

0.981205

112

191

59

0.934480

77

176

94

0.436959

42

108

25

0.980861

111

191

60

0.929760

76

175

95

0.412225

41

98

26

0.980389

110

191

61

0.922681

75

174

96

0.388792

40

105

27

0.979916

109

191

62

0.914277

74

169

97

0.367192

39

104

28

0.979481

108

191

63

0.907348

73

170

98

0.346408

38

94

29

0.979089

107

191

64

0.900439

72

169

99

0.323746

37

90

30

0.978600

106

191

65

0.892238

71

164

100

0.286501

36

84

31

0.977906

105

191

66

0.884111

70

157

101

0.267558

35

86

32

0.977417

104

191

67

0.875621

69

152

102

0.236777

34

80

33

0.976865

103

191

68

0.866776

68

152

34

0.976224

102

191

69

0.858081

67

153

35

0.975514

101

191

70

0.849840

66

148

532

Y.-C. Liang, M.-H. Lo

Appendix D

Table 9 MOVNS results of Problem III Reliability Cost Weight 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0.998064 0.998062 0.998059 0.998056 0.998053 0.998042 0.998028 0.998007 0.997986 0.997983 0.997980 0.997970 0.997955 0.997934 0.997904 0.997900 0.997890 0.997876 0.997855 0.997825 0.997809 0.997798 0.997784 0.997763 0.997733 0.997707 0.997686 0.997656 0.997627 0.997592 0.997572 0.997551 0.997521 0.997491 0.997468 0.997444 0.997414 0.997378 0.997336 0.997288

130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130

295 294 293 292 291 290 289 288 287 286 285 284 283 282 281 280 279 278 277 276 275 274 273 272 271 270 269 268 267 266 265 264 263 262 261 260 259 258 257 256

Reliability Cost Weight 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

0.997258 0.997218 0.997176 0.997128 0.997079 0.997026 0.996977 0.996923 0.996842 0.996771 0.996690 0.996590 0.996527 0.996448 0.996376 0.996295 0.996217 0.996125 0.996044 0.995945 0.995839 0.995745 0.995653 0.995589 0.995518 0.995437 0.995337 0.995238 0.995166 0.995085 0.994986 0.994796 0.994621 0.994521 0.994432 0.994317 0.994126 0.993950 0.993772 0.993673

130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130

255 254 253 252 251 250 249 248 247 246 245 244 243 242 241 240 239 238 237 236 235 234 233 232 231 230 229 228 227 226 225 224 223 222 221 220 219 218 217 216

Reliability Cost Weight 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

0.993481 0.993382 0.993183 0.992968 0.992821 0.992622 0.992410 0.992112 0.991616 0.991394 0.991182 0.990961 0.990710 0.990548 0.990350 0.990129 0.989831 0.989336 0.989106 0.988904 0.988674 0.988377 0.987882 0.987338 0.986811 0.986416 0.985922 0.985378 0.984688 0.984176 0.983505 0.982994 0.982256 0.981518 0.981027 0.980290 0.979505 0.978400 0.977596 0.976690

130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 129 130 130 129 130 129 128 126 125 126 124

215 214 213 212 211 210 209 208 207 206 205 204 203 202 201 200 199 198 197 196 195 194 193 192 191 190 189 188 187 186 185 184 183 182 181 180 179 178 177 176

Multi-objective redundancy allocation optimization using a variable

533

Table 9 (Continued) Reliability Cost Weight

Reliability Cost Weight

Reliability Cost Weight

121

0.975708

125

175

161

0.903764

90

135

201

0.634040

70

95

122

0.974926

123

174

162

0.901651

89

134

202

0.620831

68

94

123

0.973827

122

173

163

0.897098

88

133

203

0.606428

67

93

124

0.973027

123

172

164

0.892354

89

132

204

0.593794

65

92

125

0.971929

122

171

165

0.887847

88

131

205

0.587074

66

91

126

0.970760

120

170

166

0.884349

84

130

206

0.574843

64

90

127

0.969291

121

169

167

0.879882

83

129

207

0.558348

64

89

128

0.968125

119

168

168

0.875230

84

128

208

0.548667

64

88

129

0.966335

118

167

169

0.870809

83

127

209

0.540528

63

87

130

0.965042

116

166

170

0.864179

82

126

210

0.529267

61

86

131

0.963712

117

165

171

0.855450

82

125

211

0.510949

60

85

132

0.962422

115

164

172

0.849861

87

124

212

0.505167

61

84

133

0.960642

114

163

173

0.845568

86

123

213

0.494642

59

83

134

0.959188

115

162

174

0.842237

82

122

214

0.476918

61

82

135

0.958035

113

161

175

0.837983

81

121

215

0.466983

59

81

136

0.955714

112

160

176

0.833552

82

120

216

0.441039

58

80

137

0.954565

110

159

177

0.829342

81

119

217

0.430124

54

79

138

0.952560

109

158

178

0.823027

80

118

218

0.414712

56

78

139

0.951118

110

157

179

0.814714

80

117

219

0.406072

54

77

140

0.949974

108

156

180

0.807118

77

116

220

0.383512

53

76

141

0.947674

107

155

181

0.800179

79

115

221

0.372543

52

75

142

0.946534

105

154

182

0.794086

78

114

222

0.351846

51

74

143

0.944270

107

153

183

0.786065

78

113

223

0.335597

52

73

144

0.943010

106

152

184

0.778736

75

112

224

0.319579

51

72

145

0.940850

104

151

185

0.770947

78

111

225

0.307887

50

71

146

0.939595

103

150

186

0.765077

77

110

226

0.290782

49

70

147

0.937162

102

149

187

0.757349

77

109

227

0.274053

47

69

148

0.934921

104

148

188

0.750287

74

108

228

0.258828

46

68

149

0.933674

103

147

189

0.742709

74

107

150

0.931534

101

146

190

0.730608

75

106

151

0.930292

100

145

191

0.725045

74

105

152

0.927467

101

144

192

0.717721

74

104

153

0.925342

99

143

193

0.711030

71

103

154

0.924108

98

142

194

0.703847

71

102

155

0.920494

97

141

195

0.689628

70

101

156

0.918385

95

140

196

0.682519

72

100

157

0.917160

94

139

197

0.675409

70

99

158

0.912528

93

138

198

0.661765

69

98

159

0.910395

92

137

199

0.654942

71

97

160

0.908352

91

136

200

0.641298

69

96

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